3.5.69 \(\int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [469]

Optimal. Leaf size=155 \[ \frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {2 a^3 \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^2 d}+\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]

[Out]

(a^2-b^2)*x/(a^2+b^2)^2+2*a*b*ln(cos(d*x+c))/(a^2+b^2)^2/d-2*a^3*(a^2+2*b^2)*ln(a+b*tan(d*x+c))/b^3/(a^2+b^2)^
2/d+(2*a^2+b^2)*tan(d*x+c)/b^2/(a^2+b^2)/d-a^2*tan(d*x+c)^2/b/(a^2+b^2)/d/(a+b*tan(d*x+c))

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Rubi [A]
time = 0.21, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3646, 3728, 3707, 3698, 31, 3556} \begin {gather*} -\frac {a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 d \left (a^2+b^2\right )}+\frac {2 a b \log (\cos (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2}-\frac {2 a^3 \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]

[Out]

((a^2 - b^2)*x)/(a^2 + b^2)^2 + (2*a*b*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) - (2*a^3*(a^2 + 2*b^2)*Log[a + b*T
an[c + d*x]])/(b^3*(a^2 + b^2)^2*d) + ((2*a^2 + b^2)*Tan[c + d*x])/(b^2*(a^2 + b^2)*d) - (a^2*Tan[c + d*x]^2)/
(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3698

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[
A/(b*f), Subst[Int[(a + x)^m, x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]

Rule 3707

Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/((a_.) + (b_.)*tan[(e_.) + (f_.)*
(x_)]), x_Symbol] :> Simp[(a*A + b*B - a*C)*(x/(a^2 + b^2)), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2), I
nt[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Dist[(A*b - a*B - b*C)/(a^2 + b^2), Int[Tan[e + f*x], x
], x]) /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a
*B - b*C, 0]

Rule 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rubi steps

\begin {align*} \int \frac {\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=-\frac {a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\tan (c+d x) \left (2 a^2-a b \tan (c+d x)+\left (2 a^2+b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )}\\ &=\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {-a \left (2 a^2+b^2\right )-b^3 \tan (c+d x)-2 a \left (a^2+b^2\right ) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^2 \left (a^2+b^2\right )}\\ &=\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {(2 a b) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (2 a^3 \left (a^2+2 b^2\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^2 \left (a^2+b^2\right )^2}\\ &=\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (2 a^3 \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^3 \left (a^2+b^2\right )^2 d}\\ &=\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {2 a^3 \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^2 d}+\frac {\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.78, size = 329, normalized size = 2.12 \begin {gather*} \frac {a \left ((a+i b)^2 \left (-2 i a^3-4 a^2 b+2 i a b^2+b^3\right ) (c+d x)+2 a \left (a^2+b^2\right )^2 \log (\cos (c+d x))-a^3 \left (a^2+2 b^2\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )\right )+b \left (2 a^5+3 a^3 b^2+a b^4-2 i a^5 c-4 i a^3 b^2 c+a^2 b^3 c-b^5 c-2 i a^5 d x-4 i a^3 b^2 d x+a^2 b^3 d x-b^5 d x+2 a \left (a^2+b^2\right )^2 \log (\cos (c+d x))-a^3 \left (a^2+2 b^2\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )\right ) \tan (c+d x)+b^2 \left (a^2+b^2\right )^2 \tan ^2(c+d x)+2 i a^3 \left (a^2+2 b^2\right ) \text {ArcTan}(\tan (c+d x)) (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]

[Out]

(a*((a + I*b)^2*((-2*I)*a^3 - 4*a^2*b + (2*I)*a*b^2 + b^3)*(c + d*x) + 2*a*(a^2 + b^2)^2*Log[Cos[c + d*x]] - a
^3*(a^2 + 2*b^2)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]) + b*(2*a^5 + 3*a^3*b^2 + a*b^4 - (2*I)*a^5*c - (4*I
)*a^3*b^2*c + a^2*b^3*c - b^5*c - (2*I)*a^5*d*x - (4*I)*a^3*b^2*d*x + a^2*b^3*d*x - b^5*d*x + 2*a*(a^2 + b^2)^
2*Log[Cos[c + d*x]] - a^3*(a^2 + 2*b^2)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])*Tan[c + d*x] + b^2*(a^2 + b^
2)^2*Tan[c + d*x]^2 + (2*I)*a^3*(a^2 + 2*b^2)*ArcTan[Tan[c + d*x]]*(a + b*Tan[c + d*x]))/(b^3*(a^2 + b^2)^2*d*
(a + b*Tan[c + d*x]))

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Maple [A]
time = 0.16, size = 125, normalized size = 0.81

method	result	size
derivativedivides	\(\frac {\frac {\tan \left (d x +c \right )}{b^{2}}+\frac {-a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {a^{4}}{b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 a^{3} \left (a^{2}+2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(125\)
default	\(\frac {\frac {\tan \left (d x +c \right )}{b^{2}}+\frac {-a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {a^{4}}{b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 a^{3} \left (a^{2}+2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(125\)
norman	\(\frac {\frac {\tan ^{2}\left (d x +c \right )}{b d}+\frac {\left (a^{2}-b^{2}\right ) a x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {b \left (a^{2}-b^{2}\right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {\left (2 a^{3}+b^{2} a \right ) a}{d \,b^{3} \left (a^{2}+b^{2}\right )}}{a +b \tan \left (d x +c \right )}-\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 a^{3} \left (a^{2}+2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{3} d}\)	\(207\)
risch	\(-\frac {x}{2 i a b -a^{2}+b^{2}}+\frac {4 i a^{5} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{3}}+\frac {4 i a^{5} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{3} d}+\frac {8 i a^{3} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b}+\frac {8 i a^{3} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b d}-\frac {4 i a x}{b^{3}}-\frac {4 i a c}{b^{3} d}+\frac {2 i \left (-2 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{4}+2 a^{2} b^{2}+b^{4}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left (i b +a \right ) \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right ) b^{2} d}-\frac {2 a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{3} d}-\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b d}+\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{3} d}\)	\(436\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^4/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/b^2*tan(d*x+c)+1/(a^2+b^2)^2*(-a*b*ln(1+tan(d*x+c)^2)+(a^2-b^2)*arctan(tan(d*x+c)))-1/b^3*a^4/(a^2+b^2)
/(a+b*tan(d*x+c))-2/b^3*a^3*(a^2+2*b^2)/(a^2+b^2)^2*ln(a+b*tan(d*x+c)))

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Maxima [A]
time = 0.63, size = 164, normalized size = 1.06 \begin {gather*} -\frac {\frac {a^{4}}{a^{3} b^{3} + a b^{5} + {\left (a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )} + \frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{5} + 2 \, a^{3} b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}} - \frac {\tan \left (d x + c\right )}{b^{2}}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^4/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

-(a^4/(a^3*b^3 + a*b^5 + (a^2*b^4 + b^6)*tan(d*x + c)) + a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) -
(a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 2*(a^5 + 2*a^3*b^2)*log(b*tan(d*x + c) + a)/(a^4*b^3 + 2*a^2*
b^5 + b^7) - tan(d*x + c)/b^2)/d

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Fricas [A]
time = 0.88, size = 288, normalized size = 1.86 \begin {gather*} -\frac {a^{4} b^{2} - {\left (a^{3} b^{3} - a b^{5}\right )} d x - {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{6} + 2 \, a^{4} b^{2} + {\left (a^{5} b + 2 \, a^{3} b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} + {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (2 \, a^{5} b + 2 \, a^{3} b^{3} + a b^{5} + {\left (a^{2} b^{4} - b^{6}\right )} d x\right )} \tan \left (d x + c\right )}{{\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^4/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

-(a^4*b^2 - (a^3*b^3 - a*b^5)*d*x - (a^4*b^2 + 2*a^2*b^4 + b^6)*tan(d*x + c)^2 + (a^6 + 2*a^4*b^2 + (a^5*b + 2
*a^3*b^3)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) - (a^6 + 2*a
^4*b^2 + a^2*b^4 + (a^5*b + 2*a^3*b^3 + a*b^5)*tan(d*x + c))*log(1/(tan(d*x + c)^2 + 1)) - (2*a^5*b + 2*a^3*b^
3 + a*b^5 + (a^2*b^4 - b^6)*d*x)*tan(d*x + c))/((a^4*b^4 + 2*a^2*b^6 + b^8)*d*tan(d*x + c) + (a^5*b^3 + 2*a^3*
b^5 + a*b^7)*d)

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Sympy [C] Result contains complex when optimal does not.
time = 1.02, size = 2312, normalized size = 14.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**4/(a+b*tan(d*x+c))**2,x)

[Out]

Piecewise((zoo*x*tan(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((x + tan(c + d*x)**3/(3*d) - tan(c + d*x)/d)/a**
2, Eq(b, 0)), (-9*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 18*I*d
*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 9*d*x/(4*b**2*d*tan(c + d*x)
**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x
)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 8*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2
- 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 4*I*log(tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*t
an(c + d*x) - 4*b**2*d) + 4*tan(c + d*x)**3/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) +
19*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 14*I/(4*b**2*d*tan(c + d*x)*
*2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, -I*b)), (-9*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*
I*b**2*d*tan(c + d*x) - 4*b**2*d) - 18*I*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x)
- 4*b**2*d) + 9*d*x/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 4*I*log(tan(c + d*x)**2
+ 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 8*log(tan(c + d*x)**2 +
1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*I*log(tan(c + d*x)**2 + 1
)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*tan(c + d*x)**3/(4*b**2*d*tan(c + d*x)**
2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 19*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x)
- 4*b**2*d) + 14*I/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, I*b)), (x*tan(c)**4/
(a + b*tan(c))**2, Eq(d, 0)), (-2*a**6*log(a/b + tan(c + d*x))/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**
3*b**5*d + 2*a**2*b**6*d*tan(c + d*x) + a*b**7*d + b**8*d*tan(c + d*x)) - 2*a**6/(a**5*b**3*d + a**4*b**4*d*ta
n(c + d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*tan(c + d*x) + a*b**7*d + b**8*d*tan(c + d*x)) - 2*a**5*b*log(a/b +
tan(c + d*x))*tan(c + d*x)/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*tan(c + d*
x) + a*b**7*d + b**8*d*tan(c + d*x)) - 4*a**4*b**2*log(a/b + tan(c + d*x))/(a**5*b**3*d + a**4*b**4*d*tan(c +
d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*tan(c + d*x) + a*b**7*d + b**8*d*tan(c + d*x)) + a**4*b**2*tan(c + d*x)**
2/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*tan(c + d*x) + a*b**7*d + b**8*d*tan
(c + d*x)) - 3*a**4*b**2/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*tan(c + d*x)
+ a*b**7*d + b**8*d*tan(c + d*x)) + a**3*b**3*d*x/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d + 2*
a**2*b**6*d*tan(c + d*x) + a*b**7*d + b**8*d*tan(c + d*x)) - 4*a**3*b**3*log(a/b + tan(c + d*x))*tan(c + d*x)/
(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*tan(c + d*x) + a*b**7*d + b**8*d*tan(c
+ d*x)) + a**2*b**4*d*x*tan(c + d*x)/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*
tan(c + d*x) + a*b**7*d + b**8*d*tan(c + d*x)) - a**2*b**4*log(tan(c + d*x)**2 + 1)/(a**5*b**3*d + a**4*b**4*d
*tan(c + d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*tan(c + d*x) + a*b**7*d + b**8*d*tan(c + d*x)) + 2*a**2*b**4*tan
(c + d*x)**2/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*tan(c + d*x) + a*b**7*d +
b**8*d*tan(c + d*x)) - a**2*b**4/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*tan(
c + d*x) + a*b**7*d + b**8*d*tan(c + d*x)) - a*b**5*d*x/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*
d + 2*a**2*b**6*d*tan(c + d*x) + a*b**7*d + b**8*d*tan(c + d*x)) - a*b**5*log(tan(c + d*x)**2 + 1)*tan(c + d*x
)/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*tan(c + d*x) + a*b**7*d + b**8*d*tan
(c + d*x)) - b**6*d*x*tan(c + d*x)/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2*a**3*b**5*d + 2*a**2*b**6*d*tan
(c + d*x) + a*b**7*d + b**8*d*tan(c + d*x)) + b**6*tan(c + d*x)**2/(a**5*b**3*d + a**4*b**4*d*tan(c + d*x) + 2
*a**3*b**5*d + 2*a**2*b**6*d*tan(c + d*x) + a*b**7*d + b**8*d*tan(c + d*x)), True))

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Giac [A]
time = 1.17, size = 201, normalized size = 1.30 \begin {gather*} -\frac {\frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{5} + 2 \, a^{3} b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}} - \frac {2 \, a^{5} b \tan \left (d x + c\right ) + 4 \, a^{3} b^{3} \tan \left (d x + c\right ) + a^{6} + 3 \, a^{4} b^{2}}{{\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}} - \frac {\tan \left (d x + c\right )}{b^{2}}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^4/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

-(a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - (a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 2*(a^5
+ 2*a^3*b^2)*log(abs(b*tan(d*x + c) + a))/(a^4*b^3 + 2*a^2*b^5 + b^7) - (2*a^5*b*tan(d*x + c) + 4*a^3*b^3*tan
(d*x + c) + a^6 + 3*a^4*b^2)/((a^4*b^3 + 2*a^2*b^5 + b^7)*(b*tan(d*x + c) + a)) - tan(d*x + c)/b^2)/d

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Mupad [B]
time = 4.27, size = 158, normalized size = 1.02 \begin {gather*} \frac {\mathrm {tan}\left (c+d\,x\right )}{b^2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {a^4}{b\,d\,\left (\mathrm {tan}\left (c+d\,x\right )\,b^3+a\,b^2\right )\,\left (a^2+b^2\right )}-\frac {2\,a^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2+2\,b^2\right )}{b^3\,d\,{\left (a^2+b^2\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^4/(a + b*tan(c + d*x))^2,x)

[Out]

tan(c + d*x)/(b^2*d) - (log(tan(c + d*x) + 1i)*1i)/(2*d*(a*b*2i - a^2 + b^2)) - log(tan(c + d*x) - 1i)/(2*d*(2
*a*b - a^2*1i + b^2*1i)) - a^4/(b*d*(a*b^2 + b^3*tan(c + d*x))*(a^2 + b^2)) - (2*a^3*log(a + b*tan(c + d*x))*(
a^2 + 2*b^2))/(b^3*d*(a^2 + b^2)^2)

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